Representations of pure symmetric automorphism groups of RAAGs

Abstract

We study representations of the pure symmetric automorphism group $PAut(A_{\Gamma })$ of a RAAG $A_{\Gamma }$ with defining graph $\Gamma$. We first construct a homomorphism from $PAut(A_{\Gamma })$ to the direct product of a RAAG and a finite direct product of copies of $F_2\times F_2$; moreover, the image of $PAut(A_{\Gamma })$ under this homomorphism is surjective onto each factor. As a consequence, we obtain interesting actions of $PAut(A_{\Gamma })$ on non-positively curved spaces We then exhibit, for connected $\Gamma$, a RAAG which property contains $Inn(A_{\Gamma })$ and embeds as a normal subgroup of $PAut(A_{\Gamma })$. We end with a discussion of the linearity problem for $PAut(A_{\Gamma })$.

17 pages